Empirical Causal Ordering

Summary

VAR models make no a priori causal distinction between variables. Empirical methods for causal ordering — Granger causality tests, innovation accounting (impulse response functions, variance decomposition), and Cholesky decomposition — help researchers infer directional relationships from time-series data.

Granger Causality

Granger Causality

Variable $X$ Granger-causes $Y$ if past values of $X$ contain information about $Y$ beyond what is already contained in past values of $Y$ itself:

$$E[Y_{t+1}|Y_t,Y_{t-1},\dots ,X_t,X_{t-1},\dots ]\ne E[Y_{t+1}|Y_t,Y_{t-1},\dots ]$$

Test: In a bivariate VAR, test whether the block of lagged $X$ coefficients in the $Y$ equation is jointly zero:

$$Y_t=\sum _{i=1}^I{a}_i{Y}_{t-i}+\sum _{i=1}^I{b}_i{X}_{t-i}+w_{Y,t}$$

$H_0:b_1=\cdots =b_I=0$ (X does not Granger-cause Y). Test via F-test or likelihood ratio.

Important caveat: Granger causality is about predictive priority, not structural causation. It can be confounded by omitted common causes and does not imply that manipulating $X$ will change $Y$ — see Directed Acyclic Graphs and Conditional Independence Assumption.

Innovation Accounting

Two tools for understanding system dynamics in estimated VAR:

Impulse Response Function (IRF)

Impulse Response Function

The IRF traces the dynamic response of variable $Y$ to a one-time shock (innovation) in variable $X$:

$$\text{IRF}(k)=\frac{\partial Y_{t+k}}{\partial w_{X,t}}$$

For stationary systems: IRF → 0 as $k\to \infty$ (temporary effect) For I(1) systems: IRF → non-zero constant as $k\to \infty$ (multivariate persistence)

IRFs are typically plotted with 95% confidence bands computed by bootstrap or delta method. Used to visualize whether advertising shocks have persistent (Figure 7-3 right) or temporary (Figure 7-3 left) effects on sales.

Variance Decomposition (FEVD)

The forecast error variance decomposition (FEVD) decomposes the $h$-period-ahead forecast error variance of $Y$ into shares attributable to shocks in each variable:

$${\text{FEVD}}_{Y|X}(h)=\frac{{\sum }_{k=0}^{h-1}({\alpha }_{YX}^k{)}^2}{{\sum }_j{\sum }_{k=0}^{h-1}({\alpha }_{Yj}^k{)}^2}$$

At short horizons: $Y$‘s own shocks dominate. At long horizons: $X$ may explain a larger share if Granger causality is strong.

The Causal Ordering Problem

Both IRF and FEVD require orthogonalization of shocks because contemporaneous effects ($w_{Y,t}$ and $w_{X,t}$ may be correlated). The Cholesky decomposition imposes a triangular ordering:

• If advertising is ordered first, the full contemporaneous effect of advertising on sales is attributed to advertising
• If sales is ordered first, the full contemporaneous effect is attributed to sales

Sensitivity check: Good practice requires checking that key results are robust to alternative Cholesky orderings, or using alternative identification methods:

• Structural VAR (SVAR): impose economic theory to identify contemporaneous effects
• Evans-Wells (1983) method: simulate correlated shocks without requiring causal ordering

Causal Ordering in Marketing Contexts

Short-Interval Data Ordering

For weekly scanner data, a natural causal ordering is:

1. Advertising (pre-planned, set before the period)
2. Price (set by retailers, partially responding to brand decisions)
3. Sales (outcome, contemporaneously affected by advertising and price)

This ordering is defensible because consumers react to advertising and price within the measurement week, while competitor and firm decision rules operate at longer lags. Dekimpe & Hanssens (1999) argue high-frequency data makes causal ordering easier to justify.

Connection to Structural Causal Models

Granger causality is a predictive concept distinct from the structural causality in Directed Acyclic Graphs (DAGs). A Granger-causal relationship:

• May be spurious if a common cause $Z$ drives both $X$ and $Y$ with different lags
• Does not imply that an intervention on $X$ will change $Y$ (only that observing $X$ predicts $Y$)

For managerial decisions (manipulating advertising), structural identification (IV, DiD, RCT) is required — see Instrumental Variables, Differences-in-Differences.

Cross-Links

• VAR framework: Multivariate Persistence and Cointegration
• Granger causality and DAGs: Directed Acyclic Graphs
• Structural causal identification: Instrumental Variables, Conditional Independence Assumption
• Reaction functions (inter-firm Granger causality): Reaction Functions and Competitive Dynamics